which coincides with the BMS symmetry [1]}.
This choice of \(f\)  follows from the light-cone gauge conditions imposed on the metric.The transformations (REF ) and (REF ) must respect the light-cone gauge conditions on the metric \(g_{\mu \nu }\)  in order to ensure that no new degrees of freedom are introduced beyond the \(h\)  and \(\bar{h}\) . In the light-cone gauge, three metric components are set equal to zero [2]}
\(g_{--}\ =\ g_{-i}\ =\ 0\,,\quad ( i=1,2)\,,\) 
while the remaining components are parameterized as follows: \(\ g_{+-}\ =\ - e^{\phi },\ \ g_{ij}\ =\ e^{\psi } \gamma _{ij} \) ,
with \(\phi , \psi \)  real and \(\gamma _{ij}\)  real, symmetric and unimodular. The fourth gauge choice is
\(\phi = \frac{\psi }{2}\, ,\) 
fixing the time dependence of \(f\)  in terms of \(Y, \overline{Y}\) .
 Since the light-cone system is a Carroll hypersurface, the above relation can also be obtained from the conformal Carroll transformations [3]} in the null time direction \(x^+\) .
